Bulk Spin Polarization of ${\mathrm{Co}}_{(1-x)}{\mathrm{Fe}}_x{\mathbf{S}}_2$

We report on a new method to determine the degree of bulk spin polarization in single crystal ${\mathrm{Co}}_{(1-x)}{\mathrm{Fe}}_x{\mathrm{S}}_2$ by modeling magnetic Compton scattering with ab initio calculations. Spin-dependent Compton profiles were measured for ${\mathrm{CoS}}_2$ and ${\mathrm{Co}}_{0.9}{\mathrm{Fe}}_{0.1}{\mathrm{S}}_2$. The ab initio calculations were then refined by rigidly shifting the bands to provide the best fit between the calculated and experimental directional profiles for each sample. The bulk spin polarizations, $P$, corresponding to the spin-polarized density of states at the Fermi level, were then extracted from the refined calculations. The values were found to be $P=-72\pm 6\%$ and $P=18\pm 7\%$ for ${\mathrm{CoS}}_2$ and ${\mathrm{Co}}_{0.9}{\mathrm{Fe}}_{0.1}{\mathrm{S}}_2$, respectively. Furthermore, determinations of $P$ weighted by the Fermi velocity ($v_F$ or $v_F^2$) were obtained, permitting a rigorous comparison with other experimental data and highlighting the experimental dependence of $P$ on $v_F$.

Figure 1

The DOS of ${\mathrm{CoS}}_2$, where the sulfur $p$ and cobalt $d$ manifolds are depicted for the majority and minority spin states. The inset shows the band structure of the $e_g$ manifold around the Fermi level along a high symmetry path in the cubic Brillouin zone. Overall six bands composed of two in the majority (blue solid lines) and four in the minority (red dashed lines) channel cross the Fermi level.

Figure 2

The DOS in the $e_g$ manifold of (a) ${\mathrm{CoS}}_2$ and (b) ${\mathrm{Co}}_{0.9}{\mathrm{Fe}}_{0.1}{\mathrm{S}}_2$ is composed of six bands around the Fermi level. As in Fig. 1, the DOS of the majority and minority bands are blue and red, respectively. The dark and light colors signify band 1 and band 2 (or in the minority channel, the combination of bands 2–4), respectively.

Figure 3

Experimental and LMTO MCPs for (a) ${\mathrm{CoS}}_2$ and (b) ${\mathrm{Co}}_{0.9}{\mathrm{Fe}}_{0.1}{\mathrm{S}}_2$ resolved along different crystallographic directions. For clarity the profiles are offset by 0.2 and statistical errors are smaller than the symbol size. The insets of both graphs show the directional differences for each composition between [100] and [110].

Figure 4

Experimental PCAR (▲) polarization data compared to $P_{0,1,2}$ from the LMTO calculation (lines) as a function of composition in 5% steps. The tuned calculations are also shown, where the bare $P$ is also weighted by the Fermi velocity (open symbols). Note that the sign change in the PCAR data has been inferred from AMR measurements.